View the vertices as elements of $\mathbb{Z}^n$. If $a\in\mathbb{Z}^n$, define a function $f_a$ on the vertices by
$$
  f_a(x) = (-1)^{a^Tx}.
$$
This function is an eigenvectors and if $a$ has weight $w$, the eigenvalue is $n-2w$. I can make this look more combinatorial by viewing vertices (and $a$) as subsets of $\{1,\ldots,n\}$ and noting that $f_a(x)$ is determined by the parity of $a \cap x$ (abusing notation).
Different choices of $a$ give linearly independent eigenvectors, so we get the multiplicities as well as the eigenvalues.

The actual difficulty with this question is in deciding what you mean by a "graph 
theoretical proof". From where I write, linear algebra is a standard and fundamental tool in graph theory.